Learn about the Transformer Utilization Factor (TUF) in rectifiers and detailed mathematical derivation. Explore TUF values for different rectifier types like half-wave, full-wave, and bridge rectifiers.

The **Transformer Utilization Factor (TUF)** represents the efficiency of transformer utilization in a rectifier circuit. To understand TUF from a mathematical perspective, we need to derive its formula based on the relationship between the rectified DC output power and the apparent power of the transformer. Let’s go step by step through the derivation process.

**General Formula for TUF( Transformer Utilization Factor)**

The TUF is defined as the ratio of the DC power output of the rectifier (Pdc) to the apparent power rating of the transformer (Pac):

To derive the TUF expression, we need to calculate both P_{dc} and P_{ac} for the various types of rectifiers. The DC power output is calculated from the DC voltage and current, while the apparent power is based on the RMS (Root Mean Square) values of the current and voltage in the transformer.

### 1. Transformer Utilization Factor(TUF) of Half-Wave Rectifier

In a half-wave rectifier, only one-half of the AC waveform is used to produce DC output, meaning the transformer utilization is relatively low.

**DC Power Output (P _{dc})**: The average DC output voltage for a half-wave rectifier is given by:

The DC power output is:

**Apparent Power of Transformer (P _{ac})**: The apparent power is the product of the RMS voltage and RMS current. For a half-wave rectifier, the RMS values of voltage and current are:

Thus, the apparent power is:

**TUF Calculation**: Now, the TUF for a half-wave rectifier can be calculated as:

### 2. **TUF for Full-Wave Rectifier (Center-Tapped Transformer)**

In a full-wave rectifier using a center-tapped transformer, both halves of the AC waveform are utilized, leading to better transformer utilization.

**DC Power Output (P _{dc})**: The DC voltage for a full-wave rectifier is:

The DC current for a full-wave rectifier is:

And the DC power output becomes:

**Apparent Power of Transformer (P _{ac})**: The RMS voltage across each half of the transformer winding is:

Thus, the apparent power of both secondary winding is:

**TUF Calculation**:

**The TUF for a full-wave rectifier considering secondary winding apparent power:**

**The TUF for a full-wave rectifier considering primary winding apparent power:**

**Apparent Power of the Transformer (P _{ac})**

**at its primary side**

The apparent power of primary winding is:

**TUF of the center tapped full wave rectifier at primary winding of the transformer.**

Now, we can take the average of TUF at the primary and secondary winding of the transformer to calculate the TUF of the center-tapped full wave rectifier.

### 3. **TUF of Full-Wave Bridge Rectifier**

The full-wave bridge rectifier does not require a center-tapped transformer, and it utilizes both halves of the AC waveform even more efficiently.

**DC Power Output (P _{dc})**:

The DC voltage is the same as for a center-tapped full-wave rectifier.

The DC current for a full-wave rectifier is:

And the DC power output becomes:

**Apparent Power of the Transformer (P _{ac})**

**TUF Calculation**: The TUF for the bridge rectifier is:

**Summary: TUF of Different Rectifier Configuration**

S.N, | Rectifier Type | TUF Value |

1. | Half Wave Rectifier | 0.286 |

2. | Center tapped full wave Rectifier | 0.692 |

3. | Bridge Rectifier | 0.811 |